Cousin’s Lemma

We formalize, in two different ways, that “the n-dimensional Euclidean metric space is a complete metric space” (version 1. with the results obtained in [13], [26], [25] and version 2., the results obtained in [13], [14], (registrations) [24]).With the Cantor’s theorem - in complete metric space (proof by Karol Pąk in [22]), we formalize “The Nested Intervals Theorem in 1-dimensional Euclidean metric space”.Pierre Cousin’s proof in 1892 [18] the lemma, published in 1895 [9] states that: “Soit, sur le plan YOX, une aire connexe S limitée par un contour fermé simple ou complexe; on suppose qu’à chaque point de S ou de son périmètre correspond un cercle, de rayon non nul, ayant ce point pour centre : il est alors toujours possible de subdiviser S en régions, en nombre fini et assez petites pour que chacune d’elles soit complétement intérieure au cercle correspondant à un point convenablement choisi dans S ou sur son périmètre.” (In the plane YOX let S be a connected area bounded by a closed contour, simple or complex; one supposes that at each point of S or its perimeter there is a circle, of non-zero radius, having this point as its centre; it is then always possible to subdivide S into regions, finite in number and sufficiently small for each one of them to be entirely inside a circle corresponding to a suitably chosen point in S or on its perimeter) [23].Cousin’s Lemma, used in Henstock and Kurzweil integral [29] (generalized Riemann integral), state that: “for any gauge δ, there exists at least one δ-fine tagged partition”. In the last section, we formalize this theorem. We use the suggestions given to the Cousin’s Theorem p.11 in [5] and with notations: [4], [29], [19], [28] and [12].Coghetto Roland - Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Robert G. Bartle. Return to the Riemann integral. American Mathematical Monthly, pages 625–632, 1996.Robert G. Bartle. A modern theory of integration, volume 32. American Mathematical Society Providence, 2001.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Pierre Cousin. Sur les fonctions de n variables complexes. Acta Mathematica, 19(1):1–61, 1895. doi:10.1007/BF02402869.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces – fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93–102, 1999.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577–580, 2005.Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire’s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213–219, 2006. doi:10.2478/v10037-006-0024-x.Adam Grabowski and Yatsuka Nakamura. Some properties of real maps. Formalized Mathematics, 6(4):455–459, 1997.Artur Korniłowicz. Properties of connected subsets of the real line. Formalized Mathematics, 13(2):315–323, 2005.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Bernard Maurey and Jean-Pierre Tacchi. La genèse du théorème de recouvrement de Borel. Revue d’histoire des mathématiques, 11(2):163–204, 2005.Jean Mawhin. L’éternel retour des sommes de Riemann-Stieltjes dans l’évolution du calcul intégral. Bulletin de la Société Royale des Sciences de Liège, 70(4–6):345–364, 2001.Yatsuka Nakamura and Andrzej Trybulec. A decomposition of a simple closed curves and the order of their points. Formalized Mathematics, 6(4):563–572, 1997.Robin Nittka. Conway’s games and some of their basic properties. Formalized Mathematics, 19(2):73–81, 2011. doi:10.2478/v10037-011-0013-6.Karol Pąk. Complete spaces. Formalized Mathematics, 16(1):35–43, 2008. doi:10.2478/v10037-008-0006-2.Manya Raman-Sundström. A pedagogical history of compactness. The American Mathematical Monthly, 122(7):619–635, 2015.Hideki Sakurai, Hisayoshi Kunimune, and Yasunari Shidama. Uniform boundedness principle. Formalized Mathematics, 16(1):19–21, 2008. doi:10.2478/v10037-008-0003-5.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377–380, 2003.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Lee Peng Yee. The integral à la Henstock. Scientiae Mathematicae Japonicae, 67(1): 13–21, 2008.Lee Peng Yee and Rudolf Vyborny. Integral: an easy approach after Kurzweil and Henstock, volume 14. Cambridge University Press, 2000

Propositional Linear Temporal Logic with Initial Validity Semantics

In the article [10] a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of “until” operator in a very strict version. The very strict “until” operator enables to express all other temporal operators.In this article we construct a formal system for LTLB with the initial semantics [12]. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [10] and the one introduced in this article.Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in [10] and in this article can be used to carry out such verifications in Mizar [4].This work was supported by the University of Bialystok grants: BST447 Formalization of temporal logics in a proof-assistant. Application to System Verification , and BST225 Database of mathematical texts checked by computer.Faculty of Economics and Informatics, University of Białystok, Kalvariju 135, LT-08221 Vilnius, Lithuaniais work was supported by the University of Bialystok grants: BST447 Formalization of temporal logics in a proof-assistant. Application to System Verification, and BST225 Database of mathematical texts checked by computer.↩Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17. [Crossref]Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Mariusz Giero. The axiomatization of propositional linear time temporal logic. Formalized Mathematics, 19(2):113–119, 2011. doi:10.2478/v10037-011-0018-1. [Crossref]Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1): 69–72, 1999.Fred Kröger and Stephan Merz. Temporal Logic and State Systems. Springer-Verlag, 2008.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics, 8(1):133–137, 1999.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733–737, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990

Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes. We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of similar material is in [15].This work has been partially supported by the NSERC grant OGP 9207Rudnicki Piotr - University of Alberta, Edmonton, CanadaStewart Lorna - University of Alberta, Edmonton, CanadaGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.Grzegorz Bancerek. Mizar analysis of algorithms: Preliminaries. Formalized Mathematics, 15(3):87-110, 2007, doi:10.2478/v10037-007-0011-x.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Frank Harary. Graph theory. Addison-Wesley, 1969.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.J. Mycielski. Sur le coloriage des graphes. Colloquium Mathematicum, 3:161-162, 1955.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Piotr Rudnicki and Lorna Stewart. The Mycielskian of a graph. Formalized Mathematics, 19(1):27-34, 2011, doi: 10.2478/v10037-011-0005-6.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187-190, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Oswald Veblen. Analysis Situs, volume V. AMS Colloquium Publications, 193

About Graph Mappings

In this articles adjacency-preserving mappings from a graph to another are formalized in the Mizar system [7], [2]. The generality of the approach seems to be largely unpreceeded in the literature to the best of the author’s knowledge. However, the most important property defined in the article is that of two graphs being isomorphic, which has been extensively studied. Another graph decorator is introduced as [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1): 153–164, 1990.Christopher David Godsil and Gordon Royle. Algebraic graph theory. Graduate Texts in Mathematics; 207. Springer, New York, 2001. ISBN 0-387-95220-9; 0-387-95241-1.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Huckenbeck. Extremal paths in graphs: foundations, search strategies, and related topics, volume 10 of Mathematical Topics. Akademie Verlag, Berlin, 1. edition, 1997. ISBN 3-05-501658-0; 978-3-05-501658-5.Tommy R. Jensen and Bjarne Toft. Graph coloring problems. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York, 1995. ISBN 0-471-02865-7.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About supergraphs. Part I. Formalized Mathematics, 26(2):101–124, 2018. doi:10.2478/forma-2018-0009.Sebastian Koch. About supergraphs. Part II. Formalized Mathematics, 26(2):125–140, 2018. doi:10.2478/forma-2018-0010.Mike Krebs and Anthony Shaheen. Expander families and Cayley graphs: a beginners guide. Oxford University Press, Oxford, 2011. ISBN 0-19-976711-4; 978-0-19-976711-3.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.27326130

Vieta’s Formula about the Sum of Roots of Polynomials

SummaryIn the article we formalized in the Mizar system [2] the Vieta formula about the sum of roots of a polynomial anxn + an−1xn−1 + ··· + a1x + a0 defined over an algebraically closed field. The formula says that x1+x2+⋯+xn−1+xn=−an−1an , where x1, x2,…, xn are (not necessarily distinct) roots of the polynomial [12]. In the article the sum is denoted by SumRoots.Korniłowicz Artur - Institute of Informatics, University of Białystok, PolandPąk Karol - Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-817.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.Robert Milewski. Natural numbers. Formalized Mathematics, 7(1):19–22, 1998.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461–470, 2001.Piotr Rudnicki. Little Bezout theorem (factor theorem). Formalized Mathematics, 12(1): 49–58, 2004.Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.E. B. Vinberg. A Course in Algebra. American Mathematical Society, 2003. ISBN 0821834134.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.252879

Altitude, Orthocenter of a Triangle and Triangulation

We introduce the altitudes of a triangle (the cevians perpendicular to the opposite sides). Using the generalized Ceva’s Theorem, we prove the existence and uniqueness of the orthocenter of a triangle [7]. Finally, we formalize in Mizar [1] some formulas [2] to calculate distance using triangulation.Coghetto Roland - Rue de la Brasserie 5 7100 La Louvière, BelgiumGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.R. Campbell. La trigonométrie. Que sais-je? Presses universitaires de France, 1956.Wenpai Chang, Yatsuka Nakamura, and Piotr Rudnicki. Inner products and angles of complex numbers. Formalized Mathematics, 11(3):275-280, 2003.Roland Coghetto. Some facts about trigonometry and Euclidean geometry. Formalized Mathematics, 22(4):313-319, 2014. doi:10.2478/forma-2014-0031.Roland Coghetto. Morley’s trisector theorem. Formalized Mathematics, 23(2):75-79, 2015. doi:10.1515/forma-2015-0007.Roland Coghetto. Circumcenter, circumcircle and centroid of a triangle. Formalized Mathematics, 24(1):19-29, 2016. doi:10.1515/forma-2016-0002.H.S.M. Coxeter and S.L. Greitzer. Geometry Revisited. The Mathematical Association of America (Inc.), 1967.Akihiro Kubo. Lines on planes in n-dimensional Euclidean spaces. Formalized Mathematics, 13(3):389-397, 2005.Akihiro Kubo. Lines in n-dimensional Euclidean spaces. Formalized Mathematics, 11(4): 371-376, 2003.Akihiro Kubo and Yatsuka Nakamura. Angle and triangle in Euclidean topological space. Formalized Mathematics, 11(3):281-287, 2003.Marco Riccardi. Heron’s formula and Ptolemy’s theorem. Formalized Mathematics, 16 (2):97-101, 2008. doi:10.2478/v10037-008-0014-2.Boris A. Shminke. Routh’s, Menelaus’ and generalized Ceva’s theorems. Formalized Mathematics, 20(2):157-159, 2012. doi:10.2478/v10037-012-0018-9.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998

Riemann-Stieltjes Integral

In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties.In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].Narita Keiko - Hirosaki-city Aomori, JapanNakasho Kazuhisa - Akita Prefectural University Akita, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.S.L. Gupta and Nisha Rani. Fundamental Real Analysis. Vikas Pub., 1986.Einar Hille. Methods in classical and functional analysis. Addison-Wesley Publishing Co., Halsted Press, 1974.H. Kestelman. Modern theories of integration. Dover Publications, 2nd edition, 1960.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Keiichi Miyajima, Takahiro Kato, and Yasunari Shidama. Riemann integral of functions from ℝ into real normed space. Formalized Mathematics, 19(1):17-22, 2011.Daniel W. Stroock. A Concise Introduction to the Theory of Integration. Springer Science & Business Media, 1999

Topology from Neighbourhoods

Using Mizar [9], and the formal topological space structure (FMT_Space_Str) [19], we introduce the three U-FMT conditions (U-FMT filter, U-FMT with point and U-FMT local) similar to those VI, VII, VIII and VIV of the proposition 2 in [10]: If to each element x of a set X there corresponds a set B(x) of subsets of X such that the properties VI, VII, VIII and VIV are satisfied, then there is a unique topological structure on X such that, for each x ∈ X, B(x) is the set of neighborhoods of x in this topology.We present a correspondence between a topological space and a space defined with the formal topological space structure with the three U-FMT conditions called the topology from neighbourhoods. For the formalization, we were inspired by the works of Bourbaki [11] and Claude Wagschal [31].Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719–725, 1991.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93–107, 1997.Grzegorz Bancerek. Prime ideals and filters. Formalized Mathematics, 6(2):241–247, 1997.Grzegorz Bancerek. Bases and refinements of topologies. Formalized Mathematics, 7(1): 35–43, 1998.Grzegorz Bancerek, Noboru Endou, and Yuji Sakai. On the characterizations of compactness. Formalized Mathematics, 9(4):733–738, 2001.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17. [Crossref]Nicolas Bourbaki. General Topology: Chapters 1–4. Springer Science and Business Media, 2013.Nicolas Bourbaki. Topologie générale: Chapitres 1 à 4. Eléments de mathématique. Springer Science & Business Media, 2007.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Roland Coghetto. Convergent filter bases. Formalized Mathematics, 23(3):189–203, 2015. doi:10.1515/forma-2015-0016. [Crossref]Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Formalized Mathematics, 6(1):117–121, 1997.Gang Liu, Yasushi Fuwa, and Masayoshi Eguchi. Formal topological spaces. Formalized Mathematics, 9(3):537–543, 2001.Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Formalized Mathematics, 5(2):167–172, 1996.Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93–96, 1991.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233–236, 1996.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341–347, 2003.Andrzej Trybulec. Moore-Smith convergence. Formalized Mathematics, 6(2):213–225, 1997.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski – Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Josef Urban. Basic facts about inaccessible and measurable cardinals. Formalized Mathematics, 9(2):323–329, 2001.Claude Wagschal. Topologie et analyse fonctionnelle. Hermann, 1995.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215–222, 1990

Dual Lattice of ℤ-module Lattice

SummaryIn this article, we formalize in Mizar [5] the definition of dual lattice and their properties. We formally prove that a set of all dual vectors in a rational lattice has the construction of a lattice. We show that a dual basis can be calculated by elements of an inverse of the Gram Matrix. We also formalize a summation of inner products and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [20], [10] and [19].Futa Yuichi - Tokyo University of Technology, Tokyo, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-817.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.Yuichi Futa and Yasunari Shidama. Lattice of ℤ-module. Formalized Mathematics, 24 (1):49–68, 2016. doi: 10.1515/forma-2016-0005.Yuichi Futa and Yasunari Shidama. Embedded lattice and properties of Gram matrix. Formalized Mathematics, 25(1):73–86, 2017. doi: 10.1515/forma-2017-0007.Yuichi Futa and Yasunari Shidama. Divisible ℤ-modules. Formalized Mathematics, 24 (1):37–47, 2016. doi: 10.1515/forma-2016-0004.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47–59, 2012. doi: 10.2478/v10037-012-0007-z.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205–214, 2012. doi: 10.2478/v10037-012-0024-y.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of ℤ-module. Formalized Mathematics, 23(1):29–49, 2015. doi: 10.2478/forma-2015-0003.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515–534, 1982. doi: 10.1007/BF01457454.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002.Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1 (3):495–500, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877–882, 1990.Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883–885, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.25215716